Symplectic Geometry Mode Decomposition

• SGMD is a signal decomposition technique that generalizes SSA by incorporating arbitrary delay embeddings and symplectic orthogonal transforms to separate oscillatory modes.
• It replaces standard diagonal averaging with tau-slanted reconstruction, ensuring that the underlying symplectic structure is preserved in the time-domain components.
• The method connects to Hamiltonian geometry and symplectic manifolds, making it valuable for applications like machinery fault diagnostics and optimal mode separation.

Symplectic Geometry Mode Decomposition (SGMD) is a signal decomposition method for one-dimensional time sequences that combines time-delay embedding, a symplectic-geometry-aware decomposition in trajectory space, and a reconstruction step that pulls component matrices back to the original time axis. Its stated objective is to decompose a real-valued time series into data-adaptive oscillatory modes, separating transients, harmonics, modulated oscillations, and noise-sensitive structures while preserving symplectic relationships induced by the canonical form $J$. In the current arXiv literature, SGMD is understood both as a trajectory-matrix method related to Singular Spectrum Analysis (SSA) and, more broadly, as part of a family of symplectic modal decompositions connected to Hamiltonian structure, symplectic bases, and structure-preserving reductions (Zhang et al., 28 Aug 2025, Zhang et al., 12 Mar 2025, Bendokat et al., 2021, Shirafkan et al., 2021).

1. Definition and position within decomposition methods

SGMD follows the same broad pipeline as SSA: a time sequence is embedded into a trajectory matrix, the matrix is decomposed into components, and each component is mapped back to the time domain. The distinctive features of SGMD are, first, that the embedding permits an arbitrary delay $\tau \ge 1$, and second, that the matrix-space decomposition is performed with a symplectic orthogonal transform rather than a purely Euclidean SVD-based factorization. In that sense, SGMD generalizes Hankel-based SSA from unit-delay embeddings to delay-strided trajectory constructions, while importing canonical symplectic structure from Hamiltonian geometry (Zhang et al., 28 Aug 2025).

In the notation used in the recent correction and theorem papers, SGMD seeks a decomposition

$x = \sum_m x^{(m)},$

or equivalently a decomposition of the trajectory matrix into component blocks $Z_k$, whose pullbacks yield time-domain components $x^{(k)}[n]$. The 2025 pulling-back paper characterizes these as symplectic geometric components (SGCs) and presents a unified framework for decomposing both deterministic and random time sequences into multiple modes (Zhang et al., 12 Mar 2025).

Compared with empirical mode decomposition, ensemble empirical mode decomposition, or variational mode decomposition, SGMD is described in the cited literature as a linear-algebraic and geometry-grounded framework. A central theme in the 2025 work is that this framework is only mathematically consistent when the down-conversion step matches the generalized embedding; direct reuse of SSA’s classical diagonal averaging is not sufficient once $\tau>1$ or 0-based indexing is used (Zhang et al., 12 Mar 2025).

2. Trajectory-space embedding and the role of delay

For a length-$N$ sequence, SGMD forms a trajectory matrix with embedding dimension $d$ and delay $\tau$, where

$m = N - (d-1)\tau.$

In type-0 indexing, used by languages such as C, C++, Python, and Rust, the unified embedding is

$\tau \ge 1$0

In type-1 indexing, used by MATLAB, Octave, and Fortran, it is

$\tau \ge 1$1

The correction paper expresses both cases through a single embedding operator $\tau \ge 1$2, with a switch $\tau \ge 1$3 encoding the indexing convention (Zhang et al., 28 Aug 2025).

When $\tau \ge 1$4, the construction reduces to SSA’s Hankel-like embedding, in which entries representing the same sample lie on anti-diagonals. For $\tau \ge 1$5, the matrix is instead $\tau \ge 1$6-Hankel-like: equal-source entries align along lines of slope $\tau \ge 1$7 in index space. This geometric change is the source of the main theoretical issue later corrected by the pulling back theorem. The recent literature emphasizes that SGMD generalized the form of the trajectory matrix beyond SSA, but originally did not update the reconstruction principle consistently (Zhang et al., 28 Aug 2025, Zhang et al., 12 Mar 2025).

The decomposition step itself is symplectic rather than merely orthogonal. The canonical form is

$\tau \ge 1$8

and a transform $\tau \ge 1$9 is symplectic-orthogonal when

$x = \sum_m x^{(m)},$0

SGMD uses such transforms in trajectory space so that the extracted matrices $x = \sum_m x^{(m)},$1 respect the symplectic inner product induced by $x = \sum_m x^{(m)},$2. The literature frames this as better aligned with oscillatory and phase-coherent structures than a purely Euclidean decomposition (Zhang et al., 28 Aug 2025).

3. Reconstruction, diagonal averaging, and the pulling back theorem

The reconstruction problem is the most technically consequential part of current SGMD theory. Classical SSA uses the diagonal averaging principle (DAP): for $x = \sum_m x^{(m)},$3 and type-1 indexing, each reconstructed sample is the average of the entries on one anti-diagonal of a component matrix. In SGMD, however, the correct index relation is not $x = \sum_m x^{(m)},$4, but

$x = \sum_m x^{(m)},$5

with $x = \sum_m x^{(m)},$6 specifying the indexing type. Consequently, SGMD reconstruction must average along $x = \sum_m x^{(m)},$7-slanted lines, not standard anti-diagonals (Zhang et al., 28 Aug 2025).

The adjoint-based correction introduced in 2025 writes the reconstruction of a component matrix $x = \sum_m x^{(m)},$8 as

$x = \sum_m x^{(m)},$9

where $Z_k$0 is the embedding operator, $Z_k$1 is its adjoint, and $Z_k$2 is a diagonal normalization operator whose $Z_k$3-th entry equals the multiplicity $Z_k$4 of sample $Z_k$5 in the embedding. In pointwise form,

$Z_k$6

with

$Z_k$7

This is the generalized diagonal averaging principle (GDAP) (Zhang et al., 28 Aug 2025).

The unified explicit formula is

$Z_k$8

where

$Z_k$9

The theorem reduces to classical SSA DAP when $x^{(k)}[n]$0 (Zhang et al., 28 Aug 2025).

The correction paper gives a concrete counterexample with $x^{(k)}[n]$1. For type-0 indexing and $x^{(k)}[n]$2, the correct coordinates are $x^{(k)}[n]$3, $x^{(k)}[n]$4, and $x^{(k)}[n]$5, so

$x^{(k)}[n]$6

SSA’s anti-diagonal averaging does not recover that set. An analogous mismatch occurs in type-1 indexing, where the valid coordinates are $x^{(k)}[n]$7, $x^{(k)}[n]$8, and $x^{(k)}[n]$9. These examples establish that the original SGMD reconstruction was not a minor indexing inconvenience but a structural inconsistency in the inverse step (Zhang et al., 28 Aug 2025).

A common misconception is therefore that SGMD reconstruction is simply “SSA diagonal averaging applied after a different decomposition.” The current theory rejects that view. The embedding, decomposition, and pullback form a coupled construction, and the inverse must respect the same delay-strided geometry as the forward map (Zhang et al., 12 Mar 2025).

4. Symplectic structure, Hamiltonian context, and related geometric formulations

The symplectic component of SGMD is not limited to the matrix identity $\tau>1$0. Related arXiv work places symplectic mode extraction on explicit manifolds of symplectic bases and symplectic subspaces. The real symplectic Stiefel manifold is

$\tau>1$1

and the real symplectic Grassmann manifold is the manifold of $\tau>1$2-selfadjoint symplectic projectors. On these spaces, one has exact pseudo-Riemannian and Riemannian metrics, gradients, geodesics, and Cayley retractions, enabling optimization of symplectic mode bases under constraints such as

$\tau>1$3

with $\tau>1$4 (Bendokat et al., 2021).

This broader geometric framework is not identical to the time-delay SGMD correction papers, but it provides the ambient symplectic language for their decomposition step. In particular, it formalizes what it means for extracted modes or bases to preserve canonical structure and shows that symplectic mode computation can be cast as optimization over $\tau>1$5 or $\tau>1$6, with closed-form geodesics and efficient Cayley updates (Bendokat et al., 2021).

A second related direction is proper symplectic decomposition (PSD), which revisits modal analysis using Hamiltonian formalism. In that setting, phase space is $\tau>1$7 with coordinates $\tau>1$8, symplectic form

$\tau>1$9

and Hamilton’s equations

$N$0

The decomposition is obtained from intertwined eigenproblems

$N$1

with canonical normalization

$N$2

The resulting pairs $N$3 form a canonical basis, and snapshot-based estimation of a Hamiltonian operator yields a data-driven symplectic modal decomposition (Shirafkan et al., 2021).

This suggests a wider interpretation of SGMD: beyond a specific trajectory-matrix algorithm, it belongs to a structure-preserving family of decompositions in which modes are not only energy-ranked or variance-ranked, but constrained by canonical geometry. In the rotating-machinery formulation, that geometry enters through the symplectic orthogonal transform in trajectory space; in the model-reduction literature, it appears through symplectic bases, projectors, and Hamiltonian operators (Zhang et al., 28 Aug 2025, Shirafkan et al., 2021).

5. Algorithmic workflow, parameterization, and computational properties

The corrected SGMD workflow described in the 2025 literature consists of five stages. First, choose the embedding parameters $N$4, $N$5, and the indexing type $N$6, then compute $N$7. Second, construct the generalized trajectory matrix $N$8. Third, decompose $N$9 in trajectory space with a symplectic orthogonal transform to obtain component matrices $d$0. Fourth, reconstruct each component with the pulling back theorem, i.e., by applying $d$1. Fifth, group and sum components to obtain final modes (Zhang et al., 28 Aug 2025).

Grouping may be based on frequency or cyclic feature proximity, energy contribution, correlation with known fault signatures such as BPFO or BPFI, or kurtosis and cycle-entropy criteria proposed in SGMD variants. For rotating machinery, the typical pipeline includes detrending, band-pass filtering around expected defect frequencies, and normalization before decomposition; after reconstruction, one computes envelope spectra or cyclostationary features and inspects characteristic peaks and harmonics (Zhang et al., 28 Aug 2025).

Parameter selection is treated as a balance between separability, computational burden, and estimation accuracy. The correction paper states that larger $d$2 helps capture longer-period oscillations and reduce redundancy, but overly large $d$3 may undersample fast transients. Larger $d$4 increases frequency resolution and the number of copies per sample, improving separability but increasing computation. Practical heuristics in that paper are to start with $d$5 and $d$6, adjusted to sampling rate and expected fault periods, while ensuring $d$7–$d$8 for stable decomposition (Zhang et al., 28 Aug 2025).

The theorem paper gives a more explicit bound for delay selection: one may enforce $d$9, which implies

$\tau$0

Its guidance is that $\tau$1 should be selected in a limited range to balance matrix-computation efficiency against state-estimation accuracy. If $\tau$2 is small, $\tau$3 is recommended; if $\tau$4 is large, a moderate $\tau$5 can reduce matrix size while retaining enough averaging samples per reconstructed index (Zhang et al., 12 Mar 2025).

The reported complexity is standard for trajectory methods. Embedding is $\tau$6. Decomposition depends on the symplectic transform implementation and is described as typically similar to, or modestly higher than, SVD on a $\tau$7 matrix, namely $\tau$8. Reconstruction is $\tau$9 per component, because each time index aggregates between $m = N - (d-1)\tau.$0 and $m = N - (d-1)\tau.$1 entries depending on position and parameters (Zhang et al., 28 Aug 2025).

6. Applications, limitations, and current theoretical status

The most developed application context in the cited SGMD literature is rotating machinery compound fault diagnosis. SGMD is applied to vibration signals from bearings and gears under variable loads and speeds, typically sampled in the kilohertz range. The corrected pullback is expected to improve the fidelity of extracted amplitudes and modulation patterns when $m = N - (d-1)\tau.$2 or when 0-based indexing is used, thereby yielding clearer BPFO/BPFI lines, sidebands, and related spectral features (Zhang et al., 28 Aug 2025).

At the same time, the 2025 correction paper explicitly states that it focuses on theory and does not provide new quantitative results or datasets. Its practical significance is therefore inferential: by removing systematic reconstruction errors, it improves the consistency of any SGMD pipeline built on generalized embeddings. A plausible implication is that prior implementations should be audited at the reconstruction stage before their diagnostic performance is interpreted (Zhang et al., 28 Aug 2025).

The principal limitations stated in the literature remain those of delay-embedding decompositions under noise and spectral crowding. If $m = N - (d-1)\tau.$3 is too large relative to the shortest significant oscillations, information can be missed or mixed. Symplectic transforms add implementation complexity, and mode grouping still requires tuning. Mode mixing can still occur, especially under heavy noise or closely spaced frequencies, so symplecticity does not by itself guarantee complete modal separation (Zhang et al., 28 Aug 2025).

The theorem paper also distinguishes exact index correctness from robustness choices. In the noise-free case, reconstruction is uniform averaging over the valid integer solutions of the Diophantine constraint $m = N - (d-1)\tau.$4. In noisy settings, the same admissible set can be aggregated with a robust estimator such as a median or trimmed mean, written abstractly as a MeanSolver. This does not change the theorem’s index geometry; it changes only the estimator used on the correct line segment in trajectory space (Zhang et al., 12 Mar 2025).

Another recurring implementation issue is indexing. The current papers insist that $m = N - (d-1)\tau.$5 must be used for 0-based arrays and $m = N - (d-1)\tau.$6 for 1-based arrays, and that the count normalization $m = N - (d-1)\tau.$7 varies across $m = N - (d-1)\tau.$8, especially near boundaries. Mis-specifying $m = N - (d-1)\tau.$9, or using incorrect $\tau \ge 1$00/$\tau \ge 1$01 bounds, produces boundary artifacts, missed entries, and biased reconstructions (Zhang et al., 28 Aug 2025).

The present theoretical state of SGMD on arXiv is therefore defined less by new performance benchmarks than by a clarification of foundations. SGMD extends SSA by allowing arbitrary delay and symplectic trajectory-space decomposition; the pulling back theorem closes the inverse-step gap by replacing ordinary anti-diagonal averaging with adjoint-based weighted averaging along $\tau \ge 1$02-slanted lines; and related symplectic manifold and PSD research supplies a broader geometric vocabulary for understanding why symplectic modes are treated as canonical, structure-preserving objects rather than merely convenient signal components (Zhang et al., 12 Mar 2025, Bendokat et al., 2021, Shirafkan et al., 2021).